reducible, nonperiodic reducible, periodic

!?

!?

Tara Brendle and Benson Farb

!?

!!??

Tara Brendle and Benson Farb

There are numbers bigger than 2.

Order of elements

Number of elements

Genus

Brendle-Farb 2 6 g ≥ 3

Kassabov 2 4 g ≥ 7

Order of elements

Number of elements

Genus

Brendle-Farb 2 6 g ≥ 3

Kassabov 2 4 g ≥ 7

Monden 3 3 g ≥ 3

4 4 g ≥ 3

Order of elements

Number of elements

Genus

Brendle-Farb 2 6 g ≥ 3

Kassabov 2 4 g ≥ 7

Monden 3 3 g ≥ 3

4 4 g ≥ 3

Yoshihara 6 3 g ≥ 10

6 4 g ≥ 5

Obstacle:When do higher-order elements even exist in

Mod(Sg)?

Orders of torsion elements in Mod(S3):

1, 2, 3, 4, 6, 7, 8, 9, 12, 14

Number theoretic conditions for the existence of torsion elements

in Mod(Sg)

(Ashikaga and Ishizaka)

Strategy:

1) Find some order 2 elements.2) Write a Dehn twist as a product in these.3) Show that the order 2 elements generate a subgroup that puts all the Humphries curves in the same orbit.

(Luo, Brendle-Farb)

1) Find some order 2 elements.2) Write a Dehn twist as a product in these.3) Show that the order 2 elements generate a subgroup that puts all the Humphries curves in the same orbit.

Strategy:(Luo, Brendle-Farb)

(Luo)

Td = (TxTa-1)(TyTb

-1)(TzTc-1)

(Luo)

TxTa-1 = Tx(f Tx

-1f-1) = (Txf Tx-1)f-1

x a

f

Td = (TxTa-1)(TyTb

-1)(TzTc-1)

(Brendle-Farb)

(Brendle-Farb)

Strategy:

1) Find some order k elements.2) Write a Dehn twist as a product in these.3) Show that the order k elements generate a subgroup that puts all the Humphries curves in the same orbit.

genus 4genus 5

k = 5

TxTa-1 = Tx(f Tx

-1f-1) = (Txf Tx-1)f-1

Td = (TxTa-1)(TyTb

-1)(TzTc-1)

f xa

Other groups?

Sharpening these results?

Other periodic elements?

reducible elements

reducible elements

periodic elements

periodic elements

Obstacle:How can you even get a handle

on all the periodic elements?

Obstacle